Abstract 22, June (Saturday) • 10:00{10:30 Ilya M. Spitkovsky ([email protected] and [email protected]) Mathematics Program, New York University Abu Dhabi (NYUAD), UAE Title: Inverse continutiy of the numerical range map Abstract: Let A be a linear bounded operator acting on a Hilbert space H. The numerical range W (A) of A can be thought of as the image of the units sphere of H under the numerical range generating function fA : x 7! (Ax; x). This talk is devoted to continuity −1 properties of the (multivalued) inverse mapping fA . In particular, strong continuity of −1 fA on the interior of W (A) is established. Co-author: Brian Lins (Hampden-Sydney College, Virginia, USA). • 10:30{11:00 Paul Reine Kennett L. Dela Rosa ([email protected]) Department of Mathematics, Drexel University, USA Title: Location of Ritz values in the numerical range of normal matrices Abstract: In 2013, Carden and Hansen proved that fixing µ1 2 W (A)n@W (A), where A 2 3×3 C is a normal matrix with noncollinear eigenvalues, determines a unique number µ2 2 W (A) so that fµ1; µ2g forms a 2-Ritz set for A. They recognized that µ2 is the isogonal conjugate of µ1 with respect to the triangle formed by connecting the three eigenvalues of A. In this talk, we consider the analogous problem for a 4-by-4 normal matrix A. In particular, given µ1 2 W (A) in the interior of one of the quadrants formed by the diagonals of W (A), we prove that if fµ1; µ2g forms a 2-Ritz set, then µ2 lies in the convex hull of two eigenvalues of A and the two isogonal conjugates of µ1 with respect to the two triangles containing µ1. We examine how such a result can be used to understand 2-Ritz sets of n-by-n normal matrices. Co-author(s): Hugo J. Woerdeman (Department of Mathematics, Drexel University, USA). 1 • 11:15{11:45 Pan-Shun Lau ([email protected]) University of Nevada, Reno, USA Title: The decomposable numerical range of derivations Abstract: Let 1 ≤ k ≤ n be positive integers, G be a subgroup of the symmetric group of order k and χ be an irreducible character of G. The kth generalized numerical range of A 2 Cn×n associated with G and χ is defined by n o G G ∗ 2 Cn×k ∗ Wχ (A) = dχ (V AV ): V ;V V = Ik ; G where dχ is the generalized matrix function associated with G and χ. It is closely related to the decomposable numerical range of derivations. When k = 1, it reduces to the classical G numerical range of A. In the talk, we shall discuss the geometric properties of Wχ (A) such as the convexity and star-shapedness. Co-authors: Nung-Sing Sze (PolyU); Chi-Kwong Li (W&M). 2 • 11:45{12:15 Muneo Ch¯o ([email protected]) Department of Mathematics, Kanagawa University, Japan Title: Numerical ranges of Banach space operators Abstract: Let X be a complex Banach space and X ∗ be the dual space of X . For a bounded linear operator T on X , let the numerical range V (T ) of T is given by V (T ) = ff(T x):(x; f) 2 Π g; where Π is defined by Π = f (x; f) 2 X × X ∗ : kfk = f(x) = kxk = 1 g: In this talk I'll introduce properties of numerical ranges of T . References [1] M. Barra and V. M¨uller,On the essential numerical range, Acta Sci. Math. (Szeged) 71 (2005), 285-298. [2] F.F. Bonsal and J. Duncan, Numerical ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Math. Soc. Lecture Note Series. 2, 1971. [3] F.F. Bonsal and J. Duncan, Numerical ranges II, London Math. Soc. Lecture Note Series. 10, 1973. [4] M. Ch¯o,Semi-normal operators on uniformly smooth Banach spaces, Glasgow Math. J. 32 (1990), 273-276. [5] M. Ch¯o,Hyponormal operators on uniformly smooth spaces, J. Austral. Math. Soc. 50 (1991), 594-598. [6] M. Ch¯o,Hyponormal operators on uniformly convex spaces, Acta Sci. Math. (Szeged) 55 (1991), 141-147. [7] M. Ch¯oand T. Huruya, A remark on numerical range of semi-hyponormal operators, LMLA, 58 (2010), 711-714. [8] M. Ch¯oand K. Tanahashi, On conjugations for Banach spaces, Sci. Math. Jpn. 81 (2018), 37-45. [9] T. Furuta, Introduction to linear operators, Taylor and Francis, 2001. [10] S. Jung, E. Ko and J. E. Lee, On complex symmetric operators, J. Math. Anal. Appl. 406 (2013), 373-385. [11] H. Motoyoshi, Linear operators and conjugations on a Banach space, to appear in Acta Sci. Math. (Szged). 3 • 14:00{14:30 Hiroshi Nakazato ([email protected]) Department of Mathematics and Physics, Hirosaki Universit, Japan Title: Period matrix of a Riemann surface and the numerical range Abstract: We present a method to compute the period matrix of a Riemann surfacce arising as the boundary generating curve of the numerical range of a matrix. Co-author(s): This talk is based on some joint worls with Professor Mao-Ting Chien. • 14:30{15:00 Mao-Ting Chien ([email protected]) Department of Mathematics, Soochow University, Taiwan Title: Computing the diameter and width of the numerical range Abstract: The diameter(resp. width) of the numerical range of a matrix is defined to be the largest(resp. smallest) distance of two parallel lines tangent to its boundary. The boundary curve of a numerical range is called a curve of constant width if its diameter and width are equal. In this talk, we provide an algorithm for computing the diameter and width of the numerical range, formulate the diameter of the numerical range of some unitary bordering matrices, and determine the condition for the boundary of the numerical range of certain Toeplitz matrices to be a curve of constant width. Co-author(s): Hiroshi Nakazato, Jie Meng. 4 • 15:00{15:30 Masayo Fujimura ([email protected]) Department of Mathematics, National Defense Academy of Japan, Japan Title: Geometry of finite Blaschke products: pentagons and pentagrams Abstract: In this talk, I treat two types of curves induced by Blaschke product. For a Blaschke product B of degree d and λ on the unit circle, let `λ be the set of lines joining −1 each distinct two preimages in B (λ). The envelope of the family of lines f`λgλ2@D is called the interior curve associated with B. In 2002, Daepp, Gorkin, and Mortini proved the interior curve associated with a Blaschke product of degree 3 forms an ellipse. −1 While let Lλ be the set of lines tangent to the unit circle at the d preimages B (λ) and consider the trace of the intersection points of each two elements in Lλ as λ ranges over the unit circle. This trace is called the exterior curve associated with B. The exterior curve associated with a Blaschke product of degree 3 forms a non-degenerate conic. In this talk, I explain the existence of a duality-like geometrical property lies between the interior curve and the exterior curve. Using this property, I create some examples of Blaschke products whose interior curves consist of two ellipses. z2−0:47 z2−0:2 Figure 1: The envelope indicates the interior curve of B(z) = z 1−0:47z2 1−0:2z2 . The interior curve consists of two ellipses, one is inscribed in the family of pentagons and the other is inscribed in the family of pentagrams. 5 • 16:00{16:30 Patrick X. Rault ([email protected]) Department of Mathematics, University of Nebraska at Omaha, USA Title: Singularities of Base Polynomials and Gau-Wu Numbers Abstract: In 2013, Gau and Wu introduced a unitary invariant, denoted by k(A), of an n × n matrix A, which counts the maximal number of orthonormal vectors xj such that the scalar products hAxj; xji lie on the boundary of the numerical range W (A). We refer to k(A) as the Gau{Wu number of the matrix A. We write H1 and iH2 for the Hermitian and skew-Hermitian parts of A (respectively), and use them to define the base polynomial F (x; y; t) = det(xH1 + yH2 + tI). In this talk we will take an algebraic-geometric ap- proach and consider the effect of the singularities of the base curve F (x : y : t) = 0, whose dual is the boundary generating curve, to classify k(A). This continues the work of Wang and Wu classifying the Gau-Wu numbers for 3 × 3 matrices. Our focus on singularities is inspired by Chien and Nakazato, who classified W (A) for 4 × 4 unitarily irreducible A with irreducible base curve according to singularities of that curve. When A is a unitarily irreducible n×n matrix, we give necessary conditions for k(A) = 2, characterize k(A) = n, and apply these results to the case of unitarily irreducible 4 × 4 matrices. Co-author(s): Kristin A. Camenga (Houghton College), Louis Deaett (Quinnipiac Uni- versity), Tsvetanka Sendova (Michigan State University), Ilya M. Spitkovsky (New York University Abu Dhabi), Rebekah B. Johnson Yates (Houghton College) • 16:30{17:00 Michiya Mori ([email protected]) Graduate School of Mathematical Sciences, the University of Tokyo, Japan Title: Order isomorphisms of von Neumann algebras Abstract: I will consider the usual order structure of self-adjoint parts of von Neumann algebras. I will explain the general form of order isomorphisms between intervals of von Neumann algebras. In particular, I will explain that every order isomorphism between the positive cones of von Neumann algebras without commutative direct summands extends to a linear mapping.